Question

If a^2, b^2, c^2 are in A.P. Show that a+b,c+a and b+c are in H.P.​

Answer 1

Solution

we have ;

→ a² , b² ,c² are in AP

we have Show a + b , c + a , b + c are in HP

Now add ab + ca + bc on given series

→a²+ab + ca + bc , b²+ab + ca + bc ,c²+ab + ca + bc

→a(a +b) + c(a + b) ,b(b + a) +c( a +b ) ,  c( c +a ) + b (a + c )

→( a + c )( a + b ) , ( b +a )(b + c ) ,(c + a )( c + b)

now divide (a+b)(b+c)(c+a) on the term which we get

→( a + c )( a + b )/(a+b)(b+c)(c+a) , ( b +a )(b + c ) /(a+b)(b+c)(c+a)  ,(c + a )( c + b)/(a+b)(b+c)(c+a)

→1/(b+c) , 1/(a+c) , 1/(a+b)

hence proved